Monotone likelihood ratio

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May 8th 2011, 09:08 PM
andyaddition

Monotone likelihood ratio

I am trying to show that the monotone likelihood ratio for a uniform distribution [0,theta] is the maximum value (X1,...Xn). I know how to compute the monotone likelihood ratio by using fn(x given theta1)/fn(x given theta2), but then how do i show this is the maximum (X1,...Xn)?? Thanks
May 8th 2011, 09:30 PM
matheagle

do you know that the likelihood function is

${1\over \theta^n} I_{X_{(1)}>0} I_{X_{(n)}<\theta}$
May 8th 2011, 09:33 PM
andyaddition

Yes so the MLR becomes (theta2^n)/(theta1^n). But what is the step that shows that that is the maximum (X1,...Xn)??
May 8th 2011, 09:37 PM
matheagle

I'm not quite sure what your two theta's are
are you testing theta1 vs theta2?
May 8th 2011, 09:42 PM
andyaddition

Monotone likelihood ratio - Wikipedia, the free encyclopedia

It is like the formula on this page only our class uses theta1 as f(x) and theta2 as g(x)
May 8th 2011, 10:29 PM
Sambit

Suppose $\theta_2>\theta_1$ . Then the ratio becomes $\frac{\theta_1^n}{\theta_2^n}\frac{I(X_{(n)},\thet a_2)}{I(X_{(n)},\theta_1)}$ [where $I(a,b)=1 ,$ if $a\leq b$ and 0 otherwise.]

$=\frac{\theta_1^n}{\theta_2^n}\frac{1}{1}$ , if $0<X_{(n)}<\theta_1$
and $\frac{\theta_1^n}{\theta_2^n}\frac{1}{0}$ if $\theta_1<X_{(n)}<\infty$

that is $= \frac{\theta_1^n}{\theta_2^n}$ ,if $0<X_{(n)}<\theta_1$
and $= \infty$ if $\theta_1<X_{(n)}<\infty$

So as $X_{(n)}$ increases, the ratio also increases. Hence MLR in $X_{(n)}$ .
May 8th 2011, 10:46 PM
andyaddition

Thank you, but how do i show the statistic is the maximum(X1,...Xn)
May 9th 2011, 04:45 AM
Sambit

Quote:

Originally Posted by andyaddition

Thank you, but how do i show the statistic is the maximum(X1,...Xn)

How do you write the pdf (or the joint pdf) of a $U(0,\theta)$ distribution?
May 9th 2011, 06:59 AM
andyaddition

1/theta^n ??
May 9th 2011, 07:40 AM
Sambit

You are missing something. The joint pdf is given by $f(x_1,x_2,....,x_n)=\frac{1}{\theta^n},$ if $0<X_i<\theta$ $\forall i=1(1)n$ . Since all $x_i$ are less than $\theta$ , it is enough to ensure that max (X1,X2,....,Xn), which is denoted by $X_{(n)}$ , is always less than $\theta$ . Hence the joint pdf becomes $f(x_1,x_2,....,x_n)=\frac{1}{\theta^n},$ if $0<X_{(n)}<\theta$ . Now you move to the ratio.